Malcev algebra

(Redirected from Mal'cev algebra)

In mathematics, a Malcev algebra (or Maltsev algebra or MoufangLie algebra) over a field is a nonassociative algebra that is antisymmetric, so that

$xy = -yx\$

and satisfies the Malcev identity

$(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.\$

They were first defined by Anatoly Maltsev (1955).

Examples

• Any Lie algebra is a Malcev algebra.
• Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy − yx.
• The imaginary octonions form a 7-dimensional Malcev algebra by defining the Malcev product to be xy − yx.